I was wondering. I know that they practiced kendo, but it seems as though that's become much more of a sport. I'm guessing that most Japanese sword arts have some connection with the samurai, right? But what is the name of the one they practiced?

There isn't just one art. It was a combination of different arts. There is just about an art for everything in the martial arts. Kendo, kenjitsu and iaido are three arts involving the sword. There is also an art involving grappling while wearing the daisho still in there saya, the name escapes me right now. Most aikido schools have either a kenjitsu or iaido class as part of there studies.

In the times of the samurai before the Meiji Restoration there were hundreds of schools of "killing orientated" swordsmanship. The names kenjutsu, kendo etc were all used for it, allthough different terms were more popular with different people at different times.

The vague (and here quite over-generalised) distincion that kendo is a competitive sport, kenjutsu is old-style "killing orientated" swordsmanship and iaido is a mostly meditative thing is a much more modern thing. As far as I know, the swordsmanship that the samurai used is most similar to koryu kenjutsu out of all the current "systems."

Kenjutsu and iaido are just generic terms like karate and jujutsu. There are several different styles of kenjutsu and iaido. Kendo did evolve from combative practice but is pretty much just a sport now. If you want to get technical we could say iaido and kenjutsu are only sport as well since there is no telling how long it's been since there was an actual sword fight. Sword arts of any kind are still very valuable to any martial artist since weapons training will help improve your empty had skills and vise versa.